Concrete description for rank one $(\Gamma, \chi)$-theta Fock-Bargmann space in high dimension

We deal with the $(\mathbb{Z}\omega, \chi)$-theta Fock-Bargmann space consisting of holomorphic automorphic functions associated to given discrete subgroup in $\mathbb{C}^g$ of rank one and given character $\chi$. We give concrete description of its elements, it looks like a tensor product of a theta Fock-Bargmann space on the complex plane $\mathbb{C}$ and the classical Fock-Bargmann space on $\mathbb{C}^{g-1}$. Moreover, we construct an orthonormal basis and we give explicit expression of its reproducing kernel function in terms of Riemann theta function.